Math ASCII Notation Demo

Mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts.

The purpose of these pages is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity. In practice, its limits can be seen when trying to write complicated formulas (containing, for example, variables with many indexes or multiple integrals).

Despite its limitations the Math ASCII Notation has much expressive power, as can be seen from browsing through these pages.

What is an "ideal"?
Let W be a collection of sets.
W is an ideal
iff
(0) W is non-empty
(1) B,A:-W => AuB :- W
(2) B:-W and A c B => A:-W
What is a "s-ideal"?
Let W be a collection of sets.
W is a s-ideal
iff
(0) W is non-empty
(1) [ /\n:-|N A[n]:-W ] => U(n:-|N)_A[n] :- W
(2) B:-W and A c B => A:-W
Let W be a s-ring in X.
Let y : W -> [0,oo] be countably addive, y(O)=0.
Let N(y) = {BcX : \/A:-W BcA and y(A)=0}.
What can we conclude about N(y)?
N(y) is a s-ideal
page 68 in 2nd measure
Let W be a s-ring in X.
Let y : W -> [0,oo] be countably addive, y(O)=0.
Let W^ = {EcX : \/A,B:-W AcEcB and y(B\A)=0}.
What can we conclude about W^ ?
W^ is a s-ring
page 69 in 2nd measure
Let W be a ring of sets. Let y:W->|R* be additive.
Prove that if A,B:-W and AcB and y(B\A):-|R then
[ y(B):-|R <=> ??? ]
[ y(B):-|R <=> y(A):-|R ]
page 70 in 2nd measure
Let W be a ring of sets. Let y:W->|R* be additive.
Prove that if A,B:-W and AcB and y(B\A):-|R then
y(A):-|R <=> ???
y(A):-|R <=> y(B):-|R
page 70 in 2nd measure
Let x,y be real numbers.
Suppose that |x+y| <= |x+iy|.
What can we conclude?
x*y <= 0
Let x,y be real numbers.
Suppose that |x+iy| <= |x+y|.
What can we conclude?
x*y >= 0
Let W be a ring in X. Let y : W -> [0,oo] be additive.
Let A1,A2,B1,B2 be sets in W, and let E c X.
Suppose:
A1 c E c B1 and y(B1 \ A1) = 0,
A2 c E c B2 and y(B2 \ A1) = 0.
What can we conclude?
y(A1) = y(B1) = y(A2) = y(B2)
page 70 in 2nd measure
Let W be a s-ring in X.
Let y : W -> [0,oo] countably additive, y(O)=0.
Let W^ = { EcX : \/A,B:-W AcEcB and y(B\A)=0 }.
Define a countably additive function on W^, that is an extension of y.
Prove that it is countably additive.
(In this SM-collection, this function will be denoted y^.)
y^(E) = y(A) = y(B),
where A,B:-W and AcEcB and y(B\A)=0
page 71 in 2nd measure